Driving system and driving method, and exposure apparatus and exposure method

ABSTRACT

A synthetic controlled variable is obtained by obtaining a synthetic quantity using measurement results of a first and a second measuring instruments and corresponding gains (or transfer function) and synthesizing the synthetic quantity and one of the measurement results of the first and the second measuring instruments, respectively, via a high pass filter and a low pass filter. A feedback control system is structured that obtains a control input using a synthetic controlled variable and a desired value, and gives a plant the control input. This makes adding of a high pass filter for removing offset of installation position of the first and the second measuring instruments no longer necessary, and allows a driving system which controls robust driving in a high bandwidth of a plate stage regardless of bandwidth in which resonance appears to be designed.

TECHNICAL FIELD

The present invention relates to driving systems and driving methods,and exposure apparatuses and exposure methods, and more particularly toa driving system and a driving method for driving a plant by providing acontrol input, an exposure apparatus equipped with the driving system,and an exposure method that uses the driving method.

BACKGROUND ART

In a lithography process for manufacturing electronic devices(micro-devices) such as a liquid crystal display device, a semiconductordevice and the like, a projection exposure apparatus of astep-and-repeat method (a so-called stepper), a projection exposureapparatus of a step-and-scan method (a so-called scanning stepper (alsocalled a scanner)), and the like are mainly used. As for exposureapparatuses for liquid crystal display devices (liquid crystal exposureapparatus), due to the increasing size of the substrates, a scanningtype projection exposure apparatus such as the scanner is nowmainstream.

Electronic devices (micro-devices) are manufactured by forming aplurality of layers of patterns that are overlaid on a substrate (aglass plate, a wafer and the like). Therefore, in the exposureapparatus, it is necessary to accurately overlay and transfer a patternof a mask onto a pattern which is already formed in each shot area onthe substrate, that is, high overlay accuracy is required.

To achieve the high overlay accuracy, a precise and stable controltechnique of a substrate stage which moves holding the substrate will berequired. Here, in recent years, as the substrate stage, a gantry stagewhich is equipped with a carriage that moves in a scanning direction ofthe substrate at the time of scanning exposure and a substrate tablethat is supported on the carriage and moves in a non-scanning directionholding the substrate is mainly employed. In the gantry stage and thelike, resonance occurs which becomes a failure cause to an accurate andstable control of the substrate stage. Especially recently, with size ofthe substrate stage increasing, the resonance frequency tends to be low.

As a theoretical framework to structure a control system using a notchfilter, the control system being of a high frequency band including aresonance region of such substrate stages and also being robust tovariation of resonant frequency, a stage controller is known thatutilizes an advanced robust control theory represented by an H-infinitycontrol theory (for example, refer to PTL 1). In the advanced robustcontrol theory, while a sensor is added to make a plant a single-inputmultiple-output system, there are no restrictions to placement of theadded sensor, and further, a feedback controller which is also stable toa modeling error of a nominal model can be designed. However, becausedegrees of freedom when designing a controller increases in generalaccording to the plant structure, order of a weight function and thelike, a trade-off relation occurs between high bandwidth and robustnessof the feedback controller.

CITATION LIST Patent Literature

[PTL 1] Japanese Unexamined Patent Application Publication No.2002-73111

SUMMARY OF THE INVENTION Means for Solving the Problems

According to a first aspect, there is provided a first driving systemwhich drives a plant by giving control input, the system comprising: afirst measuring instrument which measures a first controlled variablerelated to a position of a first section of the plant; a secondmeasuring instrument which measures a second controlled variable relatedto a position of a second section of the plant that shows a behaviorincluding a resonance mode in opposite phase to a rigid-body mode thatthe first section shows; and a controller which obtains a thirdcontrolled variable by performing a filter processing on a measurementresult of the first and the second measuring instruments, and gives thecontrol input obtained using the third controlled variable to the plant.

According to this system, the plant can be driven precisely and in astable manner.

According to a second aspect, there is provided a first exposureapparatus that exposes an object with an energy beam and forms a patternon the object, the apparatus comprising: the first driving system of thepresent invention in which a movable body that holds the object andmoves on a predetermined plane serves as the plant.

According to this apparatus, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to a third aspect, there is provided a second driving systemwhich drives a plant by giving a control input, the system comprising: afirst measuring instrument which measures a first controlled variablerelated to a position of a first section of the plant; a secondmeasuring instrument which measures a second controlled variable relatedto a position of a second section of the plant that shows a behaviorincluding a resonance mode in opposite phase to a rigid-body mode thatthe first section shows; and a controller which obtains a thirdcontrolled variable by obtaining a plurality of synthetic quantities(X_(cn)=α_(n)X₂+β_(n)X₁ (n=1 to N)) using measurement results of thefirst and the second controlled variables (X₂, X₁) by the first and thesecond measuring instruments and a plurality of sets (N (≧2) sets) oftransfer function (α_(n), β_(n) (n=1 to N)) and performing filterprocessing on the plurality of synthetic quantities and one of themeasurement results of the first and the second measuring instruments,and gives the control input which is obtained using the third controlledvariable to the plant.

According to this system, the plant can be driven precisely and in astable manner.

According to a fourth aspect, there is provided a second exposureapparatus that exposes an object with an energy beam and forms a patternon the object, the apparatus comprising: the second driving system ofthe present invention in which a movable body that holds the object andmoves on a predetermined plane serves as the plant.

According to this apparatus, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to a fifth aspect, there is provided a third exposureapparatus that exposes an object with an energy beam and forms a patternon the object, the apparatus comprising: a movable body which has afirst movable body that moves holding the object, and a second movablebody that moves on a predetermined plane holding the first movable body;a first and a second measuring instrument that respectively measure afirst and a second controlled variable related to a position of thefirst and the second movable bodies; and a controller which obtains athird controlled variable by performing filter processing on measurementresults of the first and the second measuring instruments, and drivesthe movable body by giving the control input obtained using the thirdcontrolled variable to the movable body.

According to this apparatus, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to a sixth aspect, there is provided a first driving method inwhich a plant is driven by giving a control input, the methodcomprising: measuring a first controlled variable related to a positionof a first section of the plant and a second controlled variable relatedto a position of a second section of the plant that shows a behaviorincluding a resonance mode in opposite phase to a rigid-body mode thatfirst section shows; and obtaining a third controlled variable byperforming a filter processing on measurement results of the first andthe second controlled variables, and driving the plant by giving thecontrol input which is obtained using the third controlled variable.

According to this method, the plant can be driven precisely and in astable manner.

According to a seventh aspect, there is provided a first exposure methodthat exposes an object with an energy beam and forms a pattern on theobject, the method comprising: driving a movable body that holds theobject and moves on a predetermined plane as the plant by the firstdriving method of the present invention.

According to this method, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to an eighth aspect, there is provided a second driving methodin which a plant is driven by giving a control input, the methodcomprising: measuring a first controlled variable related to a positionof a first section the plant and a second controlled variable related toa position of a second section of the plant that shows a behaviorincluding a resonance mode in opposite phase to a rigid-body mode thatthe first section shows; and obtaining a third controlled variable byobtaining a plurality of synthetic quantities (X_(cn)=α_(n)X₂+β_(n)X₁(n=1 to N)) using measurement results of the first and the secondcontrolled variables (X₂, X₁) and a plurality of sets (N (≧2) sets) oftransfer function (α_(n), β_(n) (n=1 to N)), performing filterprocessing on the plurality of synthetic quantities and one of themeasurement results of the first and the second controlled variables,and driving the plant by giving the control input which is obtainedusing the third controlled variable.

According to this method, the plant can be driven precisely and in astable manner.

According to a ninth aspect, there is provided a second exposure methodthat exposes an object with an energy beam and forms a pattern on theobject, the method comprising: driving a movable body that holds theobject and moves on a predetermined plane as the plant by the seconddriving method of the present invention.

According to this method, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to a tenth aspect, there is provided a third exposure methodthat exposes an object with an energy beam and forms a pattern on theobject, the method comprising: measuring a first controlled variablerelated to a position of a first movable body which moves holding theobject, and a second controlled variable related to a position of asecond movable body which moves on a predetermined plane holding thefirst movable body; and driving a movable body by obtaining a thirdcontrolled variable, which is obtained performing filter processing onmeasurement results of the first and the second controlled variables,and giving to the movable body a control input obtained using the thirdcontrolled variable.

According to this method, the movable body holding the object can bedriven precisely and in a stable manner, which in turn allows exposurewith high precision to the object.

According to an eleventh aspect, there is provided a devicemanufacturing method, comprising: forming a pattern on an object usingthe second or the third exposure method of the present invention, anddeveloping the object on which the pattern is formed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a view schematically showing a structure of an exposureapparatus related to a first embodiment.

FIG. 2 is a perspective view showing a plate stage.

FIG. 3 is a block diagram showing a structure related to stage controlof the exposure apparatus.

FIG. 4 is Bode diagram showing a frequency response characteristic oftransfer function (amplitude and phase) which expresses an input-outputresponse of the plate stage in a feedback control system in asingle-input single-output system.

FIGS. 5A and 5B are Bode diagrams that respectively show a frequencyresponse characteristic of a transfer function which expresses aninput-output response of a carriage of the plate stage and a plate tablein a feedback control system in a single-input two-output system.

FIG. 6 is a block diagram expressing a feedback control system (FS-SRC)of a single-input two-output system related to the first embodiment.

FIG. 7 is a view showing an example of a dynamic model that expresses adynamical motion (translation motion) of the plate stage (a translationtwo-inertial system model).

FIG. 8A is a view showing an example of a dynamic model that expresses adynamical motion (translation motion) of the plate stage (an invertedpendulum model), and FIG. 8B is a table showing dynamic parametersincluded in the dynamic model of FIG. 8A.

FIG. 9 is a view showing an example of a dynamic model that expresses adynamical motion (translation motion) of the plate stage (atwo-resonance two-inertial spring type model).

FIG. 10 is a block diagram expressing a feedback control system (FS-SRC)of a single-input two-output system for the two-resonance two-inertialspring type model in FIG. 9.

FIG. 11 is a Bode diagram (simulation results) showing a frequencyresponse characteristic of a closed loop transfer function for afeedback control system (PID) in the conventional SISO system, afeedback control system (SRC) in the conventional SIMO system, and afeedback control system (FS-SRC) in the SIMO system of the presentembodiment, respectively.

FIG. 12 is a Nyquist diagram for each of the feedback control system(PID) in the conventional SISO system, the feedback control system (SRC)in the conventional SIMO system, and the feedback control system(FS-SRC) in the SIMO system of the present embodiment.

FIG. 13 is a block diagram expressing a feedback control system(MultiFS-SRC) of a single-input two-output system related to a secondembodiment.

FIG. 14 is a Bode diagram showing a frequency response characteristic ofa transfer function which expresses an input-output response of acarriage of a plate stage and a plate table.

FIG. 15 is a Bode diagram showing a frequency response characteristic ofa transfer function which expresses an input-output response of thecarriage of the plate stage and the plate table for feedback controlsystems (FS-SRC and MultiFS-SRC) in a SIMO system.

FIG. 16 is a Bode diagram showing a frequency response characteristic ofa closed loop transfer function for the feedback control systems (FS-SRCand MultiFS-SRC) in the SIMO system.

FIG. 17 is a Nyquist diagram for each of the feedback control systems(FS-SRC and MultiFS-SRC) in the SIMO system.

BEST MODE FOR CARRYING OUT THE INVENTION

A First Embodiment

Hereinafter, a first embodiment of the present invention will bedescribed, using FIGS. 1 to 12.

In FIG. 1, a schematic structure is shown of an exposure apparatus 110used when manufacturing flat panel displays related to the presentembodiment, such as, for example, a liquid crystal display device (aliquid crystal panel). Exposure apparatus 110 is a scanning stepper(scanner) which relatively scans a mask M on which a liquid crystaldisplay device pattern is formed and a glass plate (hereinafter,referred to as a “plate”) P held by a plate stage PST along apredetermined scanning direction (here, an X-axis direction which is alateral direction within the page surface of FIG. 1) with respect to aprojection optical system PL at the same velocity in the same direction,and transfers a pattern of mask M on plate P. In the description below,a direction in which mask M and plate P are each relatively scanned withrespect to projection optical system PL at the time of exposure will bedescribed as the X-axis direction (X direction), a direction orthogonalto the X-axis direction within a horizontal plane will be described as aY-axis direction (Y direction), and a direction orthogonal to an X-axisand a Y-axis will be described as a Z-axis direction (Z direction), androtation (tilt) directions around the X-axis, the Y-axis, and the Z-axiswill be described as θx, θy, and θz directions, respectively.

Exposure apparatus 110 is equipped with an illumination system IOP, amask stage MST which holds mask M, projection optical system PL, a bodywhich is not shown on which mask stage MST, projection optical system PLand the like are mounted, a plate stage PST which holds plate P via aplate holder PH, a control system for these parts, and the like. Thecontrol system is mainly structured by a main controller (not shown)which has overall control of each section structuring exposure apparatus110 and a stage controller 50 (refer to FIG. 3 and the like) whichoperates under the control of the main controller.

Illumination system IOP is structured similarly to an illuminationsystem disclosed in, for example, U.S. Pat. No. 5,729,331 and the like.That is, illumination system IOP irradiates light emitted from a lightsource which is not shown (for example, a mercury lamp) on mask M as anexposure illumination light (illumination light) IL, via a reflectionmirror, a dichroic mirror, a shutter, a wavelength selection filter,various kinds of lenses and the like, each of which are also not shown.As illumination light IL, for example, light such as an i-line(wavelength 365 nm), a g-line (wavelength 436 nm), or an h-line(wavelength 405 nm) (or, a synthetic light of the i-line, the g-line,and the h-line described above) is used. Further, wavelength ofillumination light IL can be switched appropriately by the wavelengthselection filter, for example, according to the resolution that isrequired.

To mask stage MST, mask M which has a circuit pattern and the likeformed on its pattern surface (the lower surface in FIG. 1) is fixed,for example, by vacuum chucking (or electrostatic suction). Mask stageMST is supported in a non-contact state (supported by levitation), via agas static bearing (for example, an air bearing) which is not shown, ona pair of mask stage guides (not shown) extending in the X-axisdirection fixed to an upper surface of a barrel surface plate which is apart of the body that is not shown. Mask stage MST is driven inpredetermined strokes in the scanning direction (the X-axis direction),for example, by a mask stage driving system MSD (not shown in FIG. 1,refer to FIG. 3) that includes a linear motor, and is also finely drivenin the Y-axis direction and the θz direction, respectively. Positioninformation (including rotation information in the θz direction) withinan XY plane of mask stage MST is measured by a mask interferometersystem 16.

Mask interferometer system 16 irradiates a measurement beam on a movablemirror (or a mirror-polished reflection surface) 15 provided on an edgeof mask stage MST, and measures the position of mask stage MST byreceiving a reflected light from movable mirror 15. The measurementresults are supplied to stage controller 50 (refer to FIG. 3), and stagecontroller 50 drives mask stage MST, via mask stage driving system MSD,based on the measurement results of mask interferometer system 16.

Projection optical system PL is supported by a part of the body which isnot shown (barrel surface plate), below mask stage MST in FIG. 1.Projection optical system PL is structured similarly to the projectionoptical system disclosed in, for example, U.S. Pat. No. 5,729,331. Thatis, projection optical system PL includes a plurality of, for example,five projection optical systems (a multi-lens projection optical system)whose projection areas of a pattern image of mask M are placed, forexample, in a staggered manner, and functions equally to a projectionoptical system that has a single image field of a rectangular shape withthe Y-axis direction being the longitudinal direction. Here, threeprojection optical systems are placed at a predetermined spacing in theY-axis direction, and the remaining two projection optical systems areset apart from the three projection optical system to the +X side andplaced at a predetermined spacing in the Y-axis direction. In thepresent embodiment, as each of the plurality of (five) projectionoptical systems, for example, an equal magnifying system is used whichis telecentric on both sides and forms an upright image. Further,hereinafter, the plurality of projection areas that are placed in astaggered manner of projection optical system PL will be referred tocollectively as an exposure area.

When the illumination area on mask M is illuminated by illuminationlight IL from illumination system IOP, by illumination light IL that haspassed through mask M, a projection image (a partially erected image) ofthe circuit pattern of mask M within the illumination area is formed viaprojection optical system PL, in an irradiation area (exposure area) ofillumination light IL which is conjugate to an illumination area onplate P whose surface is coated with a resist (sensitive agent) that isplaced on an image plane side of projection optical system PL. Then, bymask stage MST and the plate stage being synchronously driven, and maskM being relatively moved in the scanning direction (the X-axisdirection) with respect to the illumination area (illumination light IL)while plate P is relatively moved in the scanning direction (the X-axisdirection) with respect to the exposure area (illumination light IL),scanning exposure of plate P is performed, and the pattern of mask M istransferred on plate P. That is, in the present embodiment, the patternof mask M is generated on plate P by illumination system IOP andprojection optical system PL, and the pattern is formed on plate P byexposing the sensitive layer (resist layer) on plate P with illuminationlight IL.

Plate stage PST is placed below projection optical system PL (on the −Zside). Plate stage PST is equipped with a carriage 30 which moves in theX-axis direction (scanning direction), and a plate table PTB supportedon carriage 30 that moves in the non-scanning direction holding plate P.

FIG. 2 shows plate stage PST in a perspective view, along with plateinterferometer system 18 (18X, 18Y, 18X₁, and 18X₂, refer to FIG. 3).Plate table PTB, as shown in FIG. 2, consists of a plate-like memberwhich is rectangular-shaped in a planar view, and in the center of itsupper surface, plate holder PH which holds plate P by suction is fixed(not shown in FIG. 2, refer to FIG. 1). Plate table PTB is supported ona Y slider 32Y via a plurality of, for example, three support mechanisms(not shown). Each support mechanism supports plate table PTB, and alsoincludes an actuator (for example, a voice coil motor or the like) whichdrives plate table PTB in the Z-axis direction at the supporting point.By the three support mechanisms, plate table PTB is finely driven indirections of three degrees of freedom (in each of the Z-axis, the θxdirection and the θy direction) on Y slider 32Y.

Y slider 32Y has an XZ section which is an inverted U-shape, and engagesfrom above in a non-contact manner with a Y beam (Y guide) 34Y extendingin the Y-axis direction via an air bearing (not shown) and the like.Inside Y beam 34, for example, a plurality of coils are placed in theY-axis direction at a predetermined spacing, and on an inner surfaceside of Y slider 32Y, for example, a plurality of permanent magnets areplaced. By Y beam 34Y and Y slider 32Y, a moving-magnet-type Y linearmotor 36Y is structured which drives Y slider 32Y serving as a mover inthe Y-axis direction. Plate table PTB is driven in the Y-axis directionalong Y beam 34Y by Y linear motor 36Y. Incidentally, Y linear motor 36Yis not limited to the moving-magnet-type, and a moving-coil-type linearmotor can also be used.

To a lower surface at one end and the other end in the longitudinaldirection of Y beam 34Y, X sliders 32X₁ and 32X₂ are fixed. X sliders32X₁ and 32X₂ each have a YZ section which is an inverted U-shape, andare placed spaced apart in the Y-axis direction, and also engage fromabove in a non-contact manner with a pair of X guides 34X₁ and 34X₂extending in the X-axis direction via an air bearing (not shown) and thelike. X guides 34X₁ and 34X₂ are each installed, via a vibrationisolation member which is not shown (or directly) on a floor surface F.

Inside each of the X guides 34X₁ and 34X₂, for example, a plurality ofcoils are placed in the X-axis direction at a predetermined spacing, andon an inner surface of each of the X sliders 32X₁ and 32X₂, a pluralityof permanent magnets are placed. By X guide 34X₁ and X slider 32X₁, amoving-magnet-type X linear motor 36X₁ is structured which drives Xslider 32X₁ serving as a mover in the X-axis direction. Similarly, by Xguide 34X₂ and X slider 32X₂, a moving-magnet-type X linear motor 36X₂is structured which drives X slider 32X₂ serving as a mover in theX-axis direction.

Here, carriage 30 (refer to FIG. 1) is structured including the pair ofX sliders 32X₁ and 32X₂ and Y beam 34Y, and carriage 30 is driven in theX-axis direction by the pair of X linear motors 36X₁ and 36X₂. Further,by the pair of X linear motors 36X₁ and 36X₂ generating differentthrusts (driving forces), carriage 30 is driven in the θz direction bythe pair of X linear motors 36X₁ and 36X₂. Incidentally, X linear motors36X₁ and 36X₂ are not limited to the moving-magnet-type, and amoving-coil-type linear motor can also be used.

In the present embodiment, by Y linear motor 36Y, the pair of X linearmotors 36X₁ and 36X₂, and the three support mechanisms (not shown)described above, a plate stage driving system PSD (refer to FIG. 3) isstructured which drives plate table PTB in directions of six degrees offreedom (in each of the X-axis, the Y-axis, the Z-axis, the θx, the θy,and the θz directions). Stage controller 50 has control over (eachsection structuring) plate stage driving system PSD (refer to FIG. 3).

Referring back to FIG. 2, on the upper surface of plate table PTB, plateholder PH that holds plate P is fixed in the center. Further, on theupper surface of plate table PTB, at the −X end and the +Y end, amovable mirror (flat mirror) 17X having a reflection surface orthogonalto the X-axis and a movable mirror (flat mirror) 17Y having a reflectionsurface orthogonal to the Y-axis are fixed, respectively. Further, acorner cube 17X₁ is fixed on the upper surface of X slider 32X₁, and acorner cube (not shown) is fixed on the upper surface of X slider 32X₂,respectively.

A position of plate stage PST is measured by plate interferometer system18 (refer to FIG. 3). Plate interferometer system 18 includes the fourinterferometers 18X, 18Y, 18X₁ and 18X₂ shown in FIG. 2.

Interferometer 18X irradiates at least three measurement beams parallelto the X-axis on movable mirror 17X provided on plate table PTB, andmeasures the position of plate table PTB in the X-axis direction, the θzdirection, and the θY direction by receiving each of the reflectedlights. Interferometer 18Y irradiates at least two measurement beamsparallel to the Y-axis on movable mirror 17Y provided on plate tablePTB, and measures the position of plate table PTB in the Y-axisdirection and the θx direction by receiving each of the reflectedlights.

Interferometer 18X₁ irradiates a measurement beam parallel to the X-axison corner cube 17X₁ fixed on X slider 32X₁, and measures the position ofcarriage 30 in the X-axis direction (X position) by receiving thereflected light. Similarly, interferometer 18X₂ irradiates a measurementbeam parallel to the X-axis on the corner cube (not shown) fixed on Xslider 32X₂, and measures the position of carriage 30 in the X-axisdirection (X position) by receiving the reflected light.

Measurement results of each interferometer of plate interferometersystem 18 are supplied to stage controller 50 (refer to FIG. 3). Stagecontroller 50, as it will be described later on, drives plate stage PST(plate table PTB) within the XY plane using velocity of plate stage PST,via plate stage driving system PSD (to be more precise, the pair of Xlinear motors 36X₁ and 36X₂ and Y linear motor 36Y). Here, stagecontroller 50 calculates the velocity of plate stage PST by makingmeasurement results related to position from each interferometer ofplate interferometer system 18 pass a differentiator. Further, whendriving plate stage PST (plate table PTB) in the X-axis direction, as itwill be described later on, measurement results of interferometer 18X,and measurement results of at least one of interferometers 18X₁ and 18X₂are used.

Incidentally, stage controller 50 finely drives plate table PTB at leastin one of the Z-axis, the θy, and the θz directions at the time ofexposure and the like, based on detection results of a focus detectionsystem which is not shown, via plate stage driving system PSD (to bemore precise, the three support mechanisms (not shown)).

FIG. 3 shows a structure of a control system related to stage control ofexposure apparatus 110. The control system in FIG. 3 is structured, forexample, mainly of stage controller 50 which includes a microcomputerand the like.

In exposure apparatus 110, a plurality of shot areas of plate P isexposed in the following procedure, based on results of an alignmentmeasurement of the plate performed in advance (for example, EGA and thelike). That is, according to instructions from the main controller (notshown), stage controller 50 moves mask stage MST and plate stage PST totheir scanning starting positions (acceleration starting positions),while monitoring the measurement results of mask interferometer system16 and plate interferometer system 18. Then, stages MST and PST aresynchronously driven in the same direction along the X-axis direction.Accordingly, the pattern of mask M is transferred onto a shot area onplate P in the manner previously described. During the scanningexposure, stage controller 50 finely adjusts the synchronous drive(relative position and relative velocity) of mask stage MST and platestage PST, for example, according to a correction parameter. By thisoperation, the projection image of the pattern of mask M is aligned soas to overlay the pattern formed on a pre-processing layer.

When scanning exposure to one shot area is completed, stage controller50 moves (steps) plate stage PST to a scanning starting position(acceleration starting position) for the next shot area. Then, scanningexposure to the next shot area is performed. By repeating the steppingbetween shot areas of plate P and the scanning exposure to the shotareas in the manner described so far, the pattern of mask M istransferred onto all of the shot areas on plate P.

Next, a design of a driving system (a control system for controlling thedrive of plate stage PST) for plate stage PST will be described.

In the present embodiment, a driving system which drives plate stage PSTin a translation direction, as an example, in the X-axis direction willbe described. Further, for comparison, conventional art will also bebriefly described.

In the conventional art, a feedback control system (a closed loopcontrol system) which employs a single-input single-output system (SISOsystem) is structured. A case will be considered when the feedbackcontrol system of this single-input single-output system (SISO system)is applied to exposure apparatus 110. In this case, by interferometer18X, an X position (controlled variable) of plate stage PST (plate tablePTB) which serves as a plant is measured. Measurement results X aresupplied to stage controller 50. Stage controller 50 obtains a controlinput U (a driving force F that X linear motors 36X₁, 36X₂ generate, oran electric current quantity I which is to be supplied to coils of Xlinear motors 36X₁, 36X₂ or the like) using measurement results X, andsends the obtained control input U to plate stage driving system PSD.Plate stage driving system PSD, according to control input U that hasbeen received, for example, generates a driving force equivalent todriving force F, or supplies an equivalent quantity of electric currentas electric current quantity I to the coils of X linear motors 36X₁,36X₂. This controls the driving of plate stage PST.

FIG. 4 shows a Bode diagram (amplitude (gain) |P(s)| and phase arg(P(s))) that shows a frequency response characteristic of a transferfunction P (=X/U) which describes an input-output response (response ofcontrolled variable X to control input U) of plate stage PST (platetable PTB) in the feedback control system of the single-inputsingle-output system (SISO system) described above, that is, shows again diagram (view on the upper side) and a phase diagram (view on thelower side). Here, s=jω=j2πf, j=√(−1), and f is frequency. In thedrawing, for example, the solid line shows theoretical results which areobtained based on a dynamic model to be described later on, and thedashed line shows experimental results (results measured using anexperimental unit). In the experiment, controlled variable X is measuredwith respect to control input U, and by applying the results to adefinitional equation (P=X/U), the frequency response characteristic oftransfer function P is obtained.

In the frequency response characteristic of transfer function P, it canbe seen that a resonance mode (resonance behavior) appears aroundten-odd Hz. Transfer function P, as a basic behavior, monotonouslydecreases its amplitude with respect to an increase in frequency f sothat the phase is constantly maintained. These are shown in the gaindiagram and the phase diagram as a straight line with a downward slopeand a straight line with a zero slope, respectively. And, transferfunction P, as a resonance behavior, sharply increases and thendecreases its amplitude and sharply decreases and then increases itsphase in around ten-odd Hz. These are shown in the gain diagram and thephase diagram as a successive peak and trough and a trough,respectively. That is, transfer function P, in around ten-odd Hz, showsa resonance mode in opposite phase to a rigid-body mode.

The resonance mode (resonance behavior) described above appears in lowerfrequency bands due to larger exposure apparatus in recent years, andhas become a large setback for precise and stable control when drivingplate stage PST. Incidentally, in the experimental results of thefrequency response characteristic in FIG. 4, although sharp vibrationbehavior can be seen in the high frequency band (more than several tensof Hz), this will not be an issue here in particular.

To cancel out the resonance mode (resonance behavior) described aboveand to precisely and stably control the driving of plate stage PST, inaddition to interferometer 18X (a first measuring instrument) of plateinterferometer system 18, by using interferometer 18X₁ (a secondmeasuring instrument) a feedback control system which employs asingle-input two-output system (SIMO system) is structured. Here, whilethe position of carriage 30 can be measured either by interferometer18X₁ or by interferometer 18X₂, or can also be obtained by averaging themeasurement values of both interferometers, interferometer 18X₁ will beused in this case for the sake of convenience.

In the feedback control system of this single-input two-output system(SIMO system), by interferometers 18X and 18X₁, X positions (controlledvariables) X₂, X₁ of plate table PTB (a first section of the plant)structuring plate stage PST (the plant) and carriage 30 (a secondsection of the plant) are measured, respectively. These measurementresults (X₂, X₁) are supplied to stage controller 50. Stage controller50 obtains control input U (driving force F) using the measurementresults (X₂, X₁), and sends the obtained control input U to plate stagedriving system PSD. Plate stage driving system PSD (X linear motors36X₁, 36X₂) applies a driving force equivalent to driving force F tocarriage 30 (the second section), according to control input U (drivingforce F) which has been received. This drives plate stage PST.

FIG. 5A shows a Bode diagram that shows a frequency responsecharacteristic of transfer function P1 (=X₁/U) which describes aninput-output response (controlled variable X₁ with respect to controlinput U (driving force F)) of carriage 30, that is, shows a gain diagram(view on the upper side) and a phase diagram (view on the lower side).Further, FIG. 5B shows a Bode diagram that shows a frequency responsecharacteristic of transfer function P2 (=X₂/U) which describes aninput-output response (controlled variable X₂ with respect to controlinput U (driving force F)) of plate table PTB, that is, shows a gaindiagram (view on the upper side) and a phase diagram (view on the lowerside).

The frequency response characteristic of transfer function P₂ to platetable PTB (refer to FIG. 5B) shows a behavior similar to the frequencyresponse characteristic previously described (refer to FIG. 4). However,the frequency band in which the resonance behavior (resonance mode)appears shifts slightly to the high frequency side. On the contrary, thefrequency response characteristic of transfer function P1 to carriage 30shows a behavior opposing the frequency response characteristic oftransfer function P₂ (resonance mode in opposite phase), that is, showsa resonance mode in phase with the rigid-body mode. Transfer function P₁sharply decreases and then increases its amplitude to the increase offrequency f, and sharply increases and then decreases its phase. Theseare shown, in the gain diagram and the phase diagram in FIG. 5A, by asuccessive trough and peak, and a peak, respectively.

Further, an exposure apparatus that uses feedback control to a plant ofa single-input two-output system (SIMO system) is disclosed in, JapaneseUnexamined Patent Application Publication No. 2006-203113. However, theapparatus was still not sufficient because of the structure in which twooutputs are synthesized as a single output and one controller isdesigned with respect to a plant of a single-input single-output system(SISO system).

Further, in exposure apparatus 110 of the present embodiment, there isan offset in the reference positions for position measurement of platestage PST by interferometers 18X and 18X₁, that is, in the installationpositions of movable mirror 17X and corner cube 17X₁. To remove thisoffset, a high pass filter has to be connected to the controller so asto cut off controlled variable X₁ in the low frequency band. However,this arrangement causes an abnormal behavior to appear in the frequencyresponse characteristic due to the high pass filter as it will bedescribed later on, even if the feedback control system is the SIMOsystem, which prevents the designed disturbance suppressioncharacteristic from being obtained.

In exposure apparatus 110 related to the present embodiment, on buildinga feedback control system of a single-input two-output system (SIMOsystem), the second measuring instrument (interferometer 18X₁ (cornercube 17X₁)) is installed at the second section (carriage 30 (X slider32X₁) of plate stage PST which shows a behavior including a resonancemode in opposite phase to the rigid-body mode shown by the first section(plate table PTB) of plate stage PST where the first measuringinstrument (interferometer 18X (movable mirror 17X)) is installed. Bythis arrangement, a desired feedback control system can be built.

FIG. 6 shows a block diagram showing a closed loop control system(feedback control system) of a single-input two-output system (SIMOsystem) corresponding to the driving system of plate stage PST relatedto the present embodiment. The driving system corresponding to thisclosed loop control system includes, interferometers 18X and 18X₁ ofplate interferometer system 18 that measure a position (the firstcontrolled variable X₂) (in the X-axis direction) of the first section(plate table PTB) and a position (the second controlled variable X₁) (inthe X-axis direction) of the second section (carriage 30) of plate stagePST serving as the plant, respectively, a synthetic section 52 whichgenerates a synthetic controlled variable (X_(mix)) by synthesizingmeasurement results (X₂, X₁) of the first and the second controlledvariables, and stage controller 50 which computes control input U, basedon a desired value R of plate stage PST and the generation results ofsynthetic controlled variable (X_(mix)), sends the results to platestage driving system PSD, and controls the driving of plate stage PST.Here, although X positions X₂, X₁ are measured by interferometers 18X,18X₁, respectively, illustration is omitted in FIG. 6. In the blockdiagram of the closed loop control system hereinafter, illustration ofthe measuring instruments will be similarly omitted.

Here, while desired value (target trajectory), controlled variable,control input and the like are defined as a function of time, thedescription will be made using the Laplace transform of these functionsin FIG. 6 and the description using the drawing, according to thepractice in the case of explaining a control block diagram. Further,also for operational expression U (R−X_(mix)) to be described later on,the definition will be given in the form of the Laplace transform.Further, in the description below, unless it is noted in particular, thedescription will be made using the Laplace transform (in the form of theLaplace transform).

Stage controller 50 includes a target generating section 50 ₀, acontroller 50 ₁, and a subtracter 50 ₂. Incidentally, while each ofthese sections is actually realized by a microcomputer and software thatstructure stage controller 50, they can naturally be structured byhardware. Target generating section 50 ₀ generates a desired value ofplate stage PST, in this case, a target position (a desired value of aposition which changes by the minute) R, and supplies the desired valueto subtracter 50 ₂. Subtracter 50 ₂ calculates a difference betweentarget position R and synthetic controlled variable X_(mix) fromsynthetic section 52, that is, a deviation (R−X_(mix)), and supplies thedeviation to controller 50 ₁ (transfer function C). Controller 50 ₁calculates control input U=C (R−X_(mix)) by calculation (controloperation), so that deviation (R−X_(mix)) becomes zero. Here, C is atransfer function of controller 50 ₁. Transfer function is a ratioR(s)/C(s) of the Laplace transform of an input signal r(t) and an outputsignal C(t), that is, a Laplace transform function of an impulseresponse function. As is described, stage controller 50 obtains controlinput U by performing control operation expressed by operationalexpression U=C (R−X_(mix)), based on target position R and syntheticcontrolled variable X_(mix) from synthetic section 52, and gives controlinput U to plate stage PST serving as the plant. This allows plate stagePST to be driven according to control input U, and the position of platestage PST is controlled.

Synthetic section 52 includes comparators (proportional gain β, α) 52 ₁,52 ₂, an adder 52 ₃, a high pass filter 52 ₄, a low pass filter 52 ₅,and an adder 52 _(m), generates synthetic controlled variable (X_(mix))by synthesizing X position X₂ (current position) of plate table PTB(transfer function P₂) measured by interferometer 18X and X position X₁(current position) of carriage 30 (transfer function P₁) measured byinterferometer 18X₁, and supplies the synthetic controlled variable totarget generating section 50 ₀ (subtracter 50 ₂). Here, comparators(proportional gain β, α) 52 ₁, 52 ₂ multiply measurement results X₁, X₂from interferometers 18X₁, 18X by proportional gain β, α (βX₁, αX₂), andsend the multiplied measurement results to adder 52 ₃, respectively.Adder 52 ₃ generates a sum (αX₂+βX₁) of the outputs from comparators 52₁, 52 ₂, and supplies the sum to high pass filter 52 ₄. High pass filter52 ₄ and low pass filter 52 ₅ have the same cutoff frequency fc, andhigh pass filter 52 ₄ passes only frequency component F₁ (αX₂+βX₁) whichis higher than cutoff frequency fc from among signal (αX₂+βX₁) fromadder 52 ₃, whereas low pass filter 52 ₅ passes only frequency componentF2 (X₂) which is lower than cutoff frequency fc from among measurementresult X₂ from interferometer 18X, and supply the components to adder 52_(m), respectively. Adder 52 _(m) synthesizes signal F₁ (αX₂+βX₁) fromhigh pass filter 52 ₄ and signal F₂(X₂) from low pass filter 52 ₅,generates synthetic controlled variable X_(mix)=F₁ (αX₂+βX₁)+F₂(X₂), andsupplies the synthetic controlled variable to stage controller 50(subtracter 50 ₂).

As a concrete example of high pass filter 52 ₄ and low pass filter 52 ₅,a primary filter given by formula (1a) below, a secondary filter givenby formula (1b), and a quaternary filter given by formula (1c) may beincluded.

$\begin{matrix}\left\{ \begin{matrix}{F_{1} = \frac{s}{s + \omega_{f}}} \\{F_{2} = {1 - F_{1}}}\end{matrix} \right. & \left( {1a} \right) \\\left\{ \begin{matrix}{F_{1} = \frac{s^{2}}{s^{2} + {2{\zeta\omega}_{f\;}s} + \omega_{f}^{2}}} \\{F_{2} = {1 - F_{1}}}\end{matrix} \right. & \left( {1b} \right) \\\left\{ \begin{matrix}{F_{1} = \frac{s^{4} + {a_{f\; 3}s^{3}}}{s^{4} + {a_{f\; 3}s^{3}} + {a_{f\; 2}s^{2}} + {a_{f\; 1}s} + a_{f\; 0}}} \\{F_{2} = {1 - F_{1}}}\end{matrix} \right. & \left( {1c} \right)\end{matrix}$In formula (1a) and formula (1b), ω_(f=)2πfc, using cutoff frequency fc.

X_(mix) generated in the closed loop control system (feedback controlsystem) having the structure described above is plant X₂ in the lowfrequency band where there is no resonance, and in the middle and highfrequency bands where resonance exist, X_(mix) is αX₂+βX₁ which cannotbe observed with respect to resonance. Accordingly, transfercharacteristic of plate stage PST and synthetic section 52 from theinput of control input U to the output of synthetic quantity X_(mix) canbe described using an ideal rigid body model. Further, because syntheticquantity X_(mix) is equivalent to X₂ in the low frequency band, the highpass filter does not have to be connected to the controller to removethe offset between the reference positions (installation positions ofmovable mirror 17X and corner cube 17X₁) for position measurement ofinterferometers 18X and 18X₁. Furthermore, stage controller 50 can bestructured using only controller 50 ₁ which is designed based on therigid body model.

The closed loop control system (feedback control system) structured inthe manner described above will be called a frequency separation SRC(FS-SRC) type control system.

In the present embodiment, to design comparators 52 ₁, 52 ₂, that is, todecide proportional gains β, α, a dynamical motion of plate stage PST isdescribed using a simplified dynamic model (rigid model).

FIG. 7 shows a first model, a translation two-inertial system model,which describes the dynamical motion (translation motion) of plate stagePST. Plate stage PST is to be structured from two parts which are platetable PTB on which the first measuring instrument (interferometer 18X)is installed and carriage 30 on which the second measuring instruments(interferometers 18X₁, 18X₂) are installed. And motion in the X-axisdirection of these parts is expressed as a motion of two rigid bodiescoupled by a spring and a damper, or to be more precise, described as amotion of a rigid body M1 (corresponding to carriage 30) translating inthe X-axis direction by a driving force F given from a driving systemcorresponding to plate stage driving system PSD (X linear motor 36X₁,36X₂) and a rigid body M2 (corresponding to plate table PTB) which iscoupled to rigid body M1 via a spring and a damper and performstranslation motion on rigid body M1. Incidentally, the two rigid bodiescan be described, as being coupled by a spring and a damper, or as twoor more rigid bodies including the two rigid bodies to which attentionis focused on being coupled by a spring (or a spring and a damper).

Mass of the two rigid bodies (the first and the second rigid bodies)corresponding to carriage 30 and plate table PTB are to be described asM₁, M₂, respectively, rigidity coefficient and viscosity coefficient dueto friction between the first and the second rigid bodies are to be k₂,c₂, respectively, viscosity coefficient for the first rigid body is tobe c₁, and thrust acting on the second rigid body is to be F.

In the translation two-inertial system model described above, transferfunctions P₁, P₂ that express input-output response (response ofpositions X₁, X₂ to driving force F) of the first and the second rigidbodies are given as follow in the form of Laplace transform.

$\begin{matrix}{P_{1} = {\frac{X_{1}}{F} = \frac{{b_{12}s^{2}} + {b_{11}s} + b_{10}}{{a_{4}s^{4}} + {a_{3}s^{3}} + {a_{2}s^{2}} + {a_{1}s} + a_{0}}}} & \left( {2a} \right) \\{P_{2} = {\frac{X_{2}}{F} = \frac{{b_{22}s^{2}} + {b_{21}s} + b_{20}}{{a_{4}s^{4}} + {a_{3}s^{3}} + {a_{2}s^{2}} + {a_{1}s} + a_{0}}}} & \left( {2b} \right)\end{matrix}$

However,

$\begin{matrix}\left\{ \begin{matrix}{a_{4} = {M_{1}M_{2}}} \\{a_{3\;} = {{c_{1}M_{2}} + {c_{2}M_{2}} + {c_{2}M_{1}}}} \\{a_{2\;} = {{k_{2}M_{2}} + {k_{2}M_{1}} + {c_{1}c_{2}}}} \\{a_{1} = {c_{1}k_{2}}} \\{a_{0} = 0} \\{{b_{12} = M_{2}},{b_{22} = 0}} \\{{b_{11} = {b_{21} = c_{2}}},{b_{10} = {b_{20} = k_{2}}}}\end{matrix} \right. & (3)\end{matrix}$On the other hand, proportional gains α, β are decided in the followingmanner.

$\begin{matrix}\left\{ \begin{matrix}{\alpha = \frac{M_{2}}{M_{1} + M_{2}}} \\{\beta = {1 - \alpha}}\end{matrix} \right. & (4)\end{matrix}$Since deciding proportional gains α, β are similar to that of decidingin an inverted pendulum model which will be described later on, adetailed description thereabout will be omitted. When transfer functionsP₁, P₂ and proportional gains α, β are used, transfer characteristic ofX₃=αX₂+βX₁ to thrust F has a characteristic of an ideal rigid body modelas follows.

$\begin{matrix}{P_{3} = {\frac{X_{3}}{F} = \frac{1}{{\left( {M_{1} + M_{2}} \right)s^{2}} + {c_{1}s}}}} & (5)\end{matrix}$

Note that proportional gains α, β depend only on mass M₁, M₂, and do notdepend on parameters such as spring constant k₂, viscosity coefficientsc₁, c₂ that may change according to the state of plate stage PST. Thismeans that unless the resonance mode of P₁, P₂ in the closed looptransfer function is canceled out and mass M₁, M₂ of the two rigidbodies (that is, mass of carriage 30 and plate table PTB) changes, thebehavior of the closed loop transfer function remains unchangedregardless of any changes in the state of plate stage PST.

FIG. 8A shows a second model, an inverted pendulum model which expressesa dynamical motion (translation motion) of plate stage PST. Plate stagePST is to be structured from two sections, which are plate table PTB onwhich the first measuring instrument (interferometer 18X) is installedand carriage 30 on which the second measuring instrument (interferometer18X₁) is installed. And, motion in the X-axis direction of these twosections is to be expressed as motion of the two rigid bodies coupled bya spring, or to be more specific, expressed as a motion of a rigid bodyCr (corresponding to carriage 30) which is translated in the X-axisdirection by driving force F given from a driving system correspondingto plate stage driving system PSD (X linear motor 36X₁, 36X₂) and arigid body Tb (corresponding to plate table PTB) coupled via a spring ata rotation center O on rigid body Cr that rotates (in a θ_(O) direction)around rotation center O. Incidentally, the two rigid bodies can bedescribed as being coupled by a spring and a damper, or two or morerigid bodies including the two rigid bodies in focus being coupled by aspring (or a spring and a damper).

Here, X positions of rigid bodies Cr, Tb are to be expressed as X₁, X₂,respectively, mass to be expressed as M₁, M₂, respectively, moment ofinertia of rigid body Tb (regarding rotation center O) to be expressedas J, viscosity (resistance proportional to velocity of rigid body Cr)to be expressed as C, attenuation coefficient between rigid body Tb andrigid body Cr to be expressed as μ, spring constant (torsional rigiditybetween rigid body Tb and rigid body Cr) to be expressed as k, distancebetween the center of gravity of rigid body Tb and rotation center O tobe expressed as L, and separating distance in the Z-axis directionbetween the reference positions of rigid bodies Cr and Tb on X position(X₁, X₂) measurement of rigid bodies Cr and Tb to be expressed as l.Incidentally, the table in FIG. 8B shows values of these dynamicparameters. These values are decided, using a least-squares method andthe like so that model formulas expressed by formulas (2a) (2b),respectively, reproduce the frequency response characteristics oftransfer functions P₂, P₁ obtained by applying experimental results offrequency response characteristics shown in FIGS. 5A and 5B, or in otherwords, by applying to formulas (2b) (2a) experimental results of thefirst and the second controlled variables X₂, X₁ with respect to controlinput U(F).

In the inverted pendulum model described above, transfer functions P₁,P₂ that express input-output response (response of positions X₁, X₂ todriving force F) of rigid bodies Cr, Tb are given by formulas (2a) and(2b). However, it is given in the manner described below.

$\begin{matrix}\left\{ \begin{matrix}{a_{4} = {{M_{1}M_{2}L^{2}} + {\left( {M_{1} + M_{2}} \right)J}}} \\{a_{3} = {{\left( {M_{1} + M_{2}} \right)\mu} + {\left( {{M_{2}L^{2}} + J} \right)c}}} \\{a_{2} = {{\left( {M_{1} + M_{2}} \right)k} - {M_{1}M_{2}{gL}} - {M_{2}^{2}{gl}} + {\mu\; c}}} \\{a_{1} = {\left( {k - {M_{2}{gL}}} \right)c}} \\{a_{0} = 0} \\{b_{12} = {{M_{2}L^{2}} + J}} \\{b_{22} = {{M_{2}L^{2}} + J - {M_{2}{Ll}}}} \\{b_{11} = {b_{21} = \mu}} \\{b_{10} = {b_{20} = {k - {M_{2}{gL}}}}}\end{matrix} \right. & (6)\end{matrix}$

Proportional gains α, β (and transfer function C) are decided usingtransfer functions P₁, P₂ described above. For the sake of convenience,transfer functions P₁, P₂, C will be expressed in the form of fractionalexpressions P₁=N P₁/D_(P)D_(R), P₂=NP₂/D_(P)D_(R), and C=1/D_(C).However, it is as follows.N _(P1) =b ₁₂ s ² +b ₁₁ s+b ₁₀  (7a)NP ₂ =b ₂₂ s ² +b ₂₁ s+b ₂₀  (7b)D _(P) =s ² +c/(M ₁ +M ₂)s  (7c)D _(R) =a ₄ s ²+(a ₃ −a ₄ c/(M ₁ +M ₂))s+a ₁(M ₁ +M ₂)/c  (7d)

In this case, characteristic equation A_(CL) of a closed loop transferfunction with respect to a feedback control system (FIG. 6) when F₁=1,F₂=0 is given, by a numerator part of fractional expression 1+CβP₁+CαP₂.That is,A _(CL) =D _(C) D _(P) D _(R) +βN _(P1) +αN _(P2)  (8)

In characteristic equation A_(CL), α, β are decided so that thefollowing formula (9) is satisfied, using an arbitrary analyticalfunction γ.βN _(P1) +αN _(P2) =γD _(R)  (9)

From this, an open loop transfer function βP₁+αP₂=γ/D_(C)D_(P) isobtained, and pole zero cancellation is performed on poles givingresonance behaviors included in each of P₁, P₂ (that is, resonance modethat P₁, P₂ respectively show). Furthermore, D_(C), γ are decided sothat characteristic equation A_(CL) has a stable pole (in thisdescription, becomes a multiple root for the sake of convenience), thatis, so that the following formula (10) is satisfied.A _(CL)=(D _(C) D _(P)+γ)D _(R)=(s+ω ₁)(s+ω ₂) . . . (s+ω _(n))D_(R)  (10)

Next, it is decided as follows by formulas (7a) to (7d) and formula (9),so that proportional gains α, β do not include D_(R) that has asingularity (pole).

$\begin{matrix}\left\{ \begin{matrix}{\alpha = \frac{M_{2}{L/l}}{M_{1} + M_{2}}} \\{\beta = {1 - \alpha}}\end{matrix} \right. & (11)\end{matrix}$When transfer functions P₁, P₂ and proportional gains α, β are used,transfer characteristic of X₃=αX₂+βX₁ to thrust F has a characteristicof an ideal rigid body model in good approximation as follows.

$\begin{matrix}{P_{3} = {\frac{X_{3}}{F} \approx \frac{1}{{\left( {M_{1} + M_{2}} \right)s^{2}} + {cs}}}} & (12)\end{matrix}$

On deciding the remaining D_(C), γ, degrees of freedom remain to someextent. Therefore, PID controller is to be designed, for example, fromcomparators 52 ₁, 52 ₂ and controller 50 ₁. By this, D_(C)=s²+b₁s,α=b₂s²+b₃s+b₄ are obtained. However, b₁=ω₁+ω₂+ω₃+ω₄−c/(M₁+M₂),b₂=ω₁ω₂+ω₁ω₃+ω₁ω₄+ω₂ω₃+ω₂ω₄+ω₃ω₄−b₁c/(M₁+M₂),b₃=ω₁ω₂ω₃+ω₁ω₂ω₄+ω₂ω₃ω₄+ω₁ω₃ω₄, and b₄=ω₁ω₂ω₃ω₄.

Note that proportional gains α, β depend only on mass M₁, M₂ anddistances L, l, and do not depend on parameters such as spring constantk, attenuation coefficient μ, viscosity c that may change according tothe state of plate stage PST. This means that unless the resonance modeof P₁, P₂ in the closed loop transfer function is cancelled out and massM₁, M₂ (that is, mass of carriage 30 and plate table PTB) of rigidbodies Cr, Tb and distances L, l change, the behavior of the closed looptransfer function remains unchanged regardless of any changes in thestate of plate stage PST.

FIG. 9 shows a third model, a two-resonance two-inertial spring typemodel, which expresses a dynamical motion (translation motion) of platestage PST. The two-resonance two-inertial spring type model expressesplate stage PST, which performs translation motion for each of the twosections, plate table PTB on which the first measuring instrument(interferometer 18X) is installed and carriage 30 on which the secondmeasuring instrument (interferometer 18X₁) installed, as a translationmotion of two rigid bodies coupled by a spring and a damper.Incidentally, the two rigid bodies can be expressed as being coupledonly by a spring, or two or more rigid bodies including the focused tworigid bodies can be coupled by a spring and a damper (or only by aspring).

Mass of the two rigid bodies (a first and a second rigid body)corresponding to plate table PTB and carriage 30 is to be M₂, M₁,respectively, rigidity coefficient and viscosity coefficient for thefirst rigid body are to be k₀, c₀, respectively, rigidity coefficientand viscosity coefficient for the second rigid body are to be k₁, c₁,respectively, rigidity coefficient and viscosity coefficient due tofriction between the first and the second rigid bodies are to be k₂, c₂,respectively, and thrust acting on the first rigid body is to be F.

In the two-resonance two-inertial spring type model described above,transfer functions P₁, P₂ that express input-output response (responseof positions X₁, X₂ to driving force F) of the first and the secondrigid bodies are given by formulas (2a) and (2b). However, the followingexpression is applied.

$\begin{matrix}\left\{ \begin{matrix}{a_{4} = {M_{1}M_{2}}} \\{a_{3} = {{M_{1}{c\;}_{2}} + {M_{2}c_{1}} + {M_{2}c_{2}} + {M_{1}c_{0}}}} \\{a_{2} = {{M_{1}k_{2}} + {M_{2}k_{1}} + {M_{2}k_{2}} + {M_{1}k_{0}} + {c_{1}c_{2}} + {c_{0}\left( {c_{1} + c_{2}} \right)}}} \\{a_{1} = {{c_{1}k_{2}} + {c_{2}k_{1}} + {c_{0}\left( {k_{1} + k_{2}} \right)} + {\left( {c_{1} + c_{2}} \right)k_{0}}}} \\{a_{0} = {{k_{1}k_{2}} + {k_{0}\left( {k_{1} + k_{2}} \right)}}} \\{{b_{12} = 0},{b_{22} = M_{1}}} \\{{b_{11} = c_{2}},{b_{21} = {c_{1} + c_{2}}},{b_{10} = k_{2}},{b_{20} = {k_{1} + k_{2}}}}\end{matrix} \right. & (13)\end{matrix}$Incidentally, to this two-resonance two-inertial spring type model,instead of the feedback control system expressed by the block diagram inFIG. 6, a feedback control system expressed by a block diagram in FIG.10 is employed. That is, comparators (proportional gains β, α) 52 ₁, 52₂ in FIG. 6 are replaced by controllers (transfer functions β, α) 52 ₇,52 ₈ in FIG. 10. Corresponding to this, proportional gains β, α arereplaced to transfer functions β=β (s), α=α (s). For the sake ofconvenience, the transfer functions will be expressed using the samenotations as proportional gains β, α.

For the two-resonance two-inertial spring type model, transfer functionsα, β are determined in the following manner.

$\begin{matrix}\left\{ \begin{matrix}{\alpha = {\left( {{M_{2}s^{2}} + {c_{0}s} + k_{0}} \right)\frac{\omega_{np}^{2}}{s^{2} + {2{\zeta\omega}_{np}s} + \omega_{np}^{2}}{P_{2}(0)}}} \\{\beta = {\left( {{M_{1}s^{2}} + {c_{1}s} + k_{1}} \right)\frac{\omega_{np}^{2}}{s^{2} + {2{\zeta\omega}_{np}s} + \omega_{np}^{2}}{P_{2}(0)}}}\end{matrix} \right. & (14)\end{matrix}$However, P₂ is given as follows.

$\begin{matrix}{{P_{2}(0)} = \frac{k_{1} + k_{2}}{{k_{1}k_{2}} + {k_{0}\left( {k_{1} + k_{2}} \right)}}} & (15)\end{matrix}$Because deciding transfer functions α, β are similar to deciding in theinverted pendulum model previously described, details thereabout will beomitted. By this, transfer characteristic of X₃=αX₂+βX₁ with respect tothrust F has a characteristic of an ideal secondary low pass filter asfollows.

$\begin{matrix}{P_{3} = {\frac{X_{3}}{F} = {\frac{\omega_{np}^{2}}{s^{2} + {2{\zeta\omega}_{np}s} + \omega^{2}}{P_{2}(0)}}}} & (16)\end{matrix}$Here, ζ and ω_(n)p can each be designed arbitrarily, and are anattenuation ratio (damping factor) and a natural angular frequency of anideal secondary low pass filter characteristic from thrust F to X₃.

Note that transfer functions α, β do not depend on viscosity coefficientc₂ due to friction between the first and the second rigid bodies, thatis, do not depend on parameters that may change according to the statebetween plate stage PST and carriage 30 which are equivalent to thefirst and the second rigid bodies. This means that the resonance mode ofP₁, P₂ in the closed loop transfer function is cancelled out and thatthe behavior of the closed loop transfer function remains unchanged tochanges in the state between plate stage PST and carriage 30.

The inventors verified by simulation the performance of the feedbackcontrol system (FS-SRC) based on the SIMO system designed in the mannerdescribed above. Further, for comparison, the inventors also verifiedthe performance of a feedback control system (called a PID) of aconventional single-input single-output system (SISO system), made up ofa combination of a PID type controller and a notch filter (for example,refer to Japanese Unexamined Patent Application Publication No.2006-203113), and a conventional SRC type feedback control system(called an SRC) that uses the first and the second controlled variables(X₂, X₁) without performing filter synthesis.

Dynamical motion (response characteristic) of plate stage PST isreproduced using the inverted pendulum model previously described. Here,values are used of the dynamic parameter compiled in the table of FIG.8B. Further, a PID type controller was employed for the controller (suchas C) used in FS-SRC and SRC. Further, the controller was designed inthe same pole placement, along with the three feedback control systems.Further, in the SRC, to remove the offset between the reference position(installation position of movable mirror 17X and corner cube 17X₁) onposition measurement of interferometers 18X, 18X₁, a secondary high passfilter with a cutoff frequency fc=5 Hz was added to the controller ofinterferometer 18X₁ (controlled variable X₂). Filters 52 ₁, 52 ₂ ofFS-SRC are secondary filters, with a cutoff frequency of 1 Hz.

FIG. 11 shows a gain diagram which indicates a frequency responsecharacteristic of a sensitivity function (closed loop transfer function)S of FS-SRC by the SIMO system of the present embodiment. Further, forcomparison, a gain diagram is also shown which indicates a frequencyresponse characteristic of PID by the conventional SISO system and ofSRC by the conventional SIMO system. An abnormal behavior caused by thehigh pass filter appears at ten-odd Hz in the PID by the conventionalSISO system, in the SRC by the conventional SIMO system, and in theFS-SRC by the SIMO system of the present embodiment. However, it can beseen that the degree of the abnormal behavior is sufficiently smaller inthe FS-SRC by the SIMO system of the present embodiment than that of thePID by the conventional SISO system and the SRC by the conventional SIMOsystem.

Further, because the high pass filter was added to the SRC, sensitivityperformance equal to the PID cannot be obtained in the low frequencyrange even though the controller was designed in the same poleplacement. In contrast, sensitivity characteristic of the FS-SRC becomesequal to the PID in the low frequency range, and furthermore shows anideal sensitivity characteristic without any abnormal behavior beinggenerated by the resonance mode.

FIG. 12 shows a Nyquist diagram. It can be seen that both the SRC andthe FS-SRC are not affected by the resonance, and that stability marginis sufficiently large when compared to the PID.

As is described so far, according to exposure apparatus 110 related tothe present embodiment, on carriage 30 (the second section of the plant)that shows a behavior including a resonance mode in opposite phase tothe rigid-body mode that plate table PTB (the first section of theplant) on which interferometer 18X (the first measuring instrument)measuring position (the first controlled variable) X₂ of plate stage PST(plant) is installed shows, interferometer 18X₁ (the second measuringinstrument) which measures position (the second controlled variable) X₁of plate stage PST is installed. By using the first and the secondmeasuring instruments, it becomes possible to design a control drivingsystem which can control robust driving of plate stage PST in the highbandwidth.

Further, in the feedback control system of the SIMO system in thepresent embodiment, the structure is employed where filter processing isperformed on the measurement results of interferometer 18X (the firstmeasuring instrument) and interferometer 18X₁ (the second measuringinstrument) to obtain synthetic controlled variable X_(mix), syntheticcontrolled variable X_(mix) and desired value R are used to obtaincontrol input U, and the control input is given to the plant. Here,synthetic controlled variable X_(mix)=F₁ (αX₂+βX₁)+F₂(X₂) is obtained byobtaining synthetic quantity (X_(c)=αX₂+βX₁) using measurement results(X₂, X₁) of interferometer 18X (the first measuring instrument) andinterferometer 18X₁ (the second measuring instrument) and gains (ortransfer functions) (α, β) corresponding to the measurement results, andsynthesizing synthetic quantity (X_(c)) and measurement results (X₂, X₁)of one of the first and the second measuring instruments, respectively,via high pass filter (F₁) and low pass filter (F₂) having the samecutoff frequency as the high pass filter.

In the conventional feedback control system of the SIMO system, becausethere is an offset in the reference position on X position measurementof plate stage PST by the first and the second measuring instruments(interferometers 18X, 18X₁), that is, an offset in the installationpositions of movable mirror 17X and corner cube 17X₁, to remove thisoffset, a high pass filter has to be connected to cut the controlledvariable in the low frequency band. However, when the bandwidth in whichthe resonance appears is low and overlaps the frequency band where thecontrolled variable is cut, because the signal for self-cancelling (tocancel the resonance mode of P₁ by the resonance mode of P₂) theresonance mode also becomes cut, this may end up reducing the controlaccuracy. Whereas, in the feedback control system of the SIMO system inthe present embodiment, by the structure described above, becausesynthetic controlled variable X_(mix) is X₂ of the plant in the lowfrequency band where there is no resonance and becomes αX₂+βX₁ which isunobservable to resonance in the middle and high frequency band wherethere is resonance, the high pass filter to remove the offset does nothave to be connected to the controller, and furthermore, stagecontroller 50 can be structured using only controller 50 ₁ which isdesigned based on the rigid body model. This allows a driving systemwhich controls robust driving of plate stage PST in the high bandwidthto be designed, regardless of the bandwidth where the resonance appears.

Further, gains (or transfer functions) β, α are decided so that thepoles corresponding to the resonance modes included in each of thetransfer functions P₂, P₁ that express the responses of the first andthe second sections (plate table PTB and carriage 30) of plate stage PSTare cancelled out in the open loop transfer function βP₁+αP₂.Furthermore, specific forms of transfer functions P₂, P₁ are given usinga dynamic model (rigid model) which expresses the motion of the firstand the second sections as a motion of the two rigid bodies coupled by aspring. This cancels out the resonance behaviors (resonance modes) ofP₂, P₁ in the closed loop transfer function (the resonance mode of theplant is canceled out by a linear sum of an anti-resonant mode of P₂ andP₁), making it possible to design a driving system which controls robustdriving of plate stage PST to any change of state.

Further, because exposure apparatus 110 related to the presentembodiment is equipped with a driving system of plate stage PST designedin the manner described above, it becomes possible to accurately andstably drive plate stage PST, which allows exposure accuracy, or inother words, overlay accuracy to be improved.

Incidentally, in the embodiment above, while stabilization of platestage PST was to be improved by deciding transfer functions (α, β) sothat the poles corresponding to the resonance modes included in each ofthe transfer functions P₂, P₁ corresponding to the first and the secondsections are canceled out in the open loop transfer function βP₁+αP₂,the present invention is not limited to this. For example, transferfunctions (α, β) may be obtained without canceling out the polescorresponding to the resonance modes included in each of the transferfunctions P₂, P₁, by improving the damping effect of the plate stage tostabilize the resonance mode. In the present embodiment, in the Nyquistdiagram, the size and the orientation of the circle which shows theresonance mode can be set freely within the respective characteristicrange of transfer functions P₂, P₁. As a guideline for stabilization,for example, transfer functions (α, β) may be set so that the circleshowing the resonance mode is in a state positioned almost on the firstquadrant and the fourth quadrant (right-half plane), or in other words,in a state barely positioned on the second quadrant and the thirdquadrant (left-half plane).

Further, in the feedback control system of the SIMO system in presentembodiment, while the synthetic quantity (X_(c)=αX₂+βX₁) obtained fromthe measurement results (X₂, X₁) of interferometer 18X (the firstmeasuring instrument) and interferometer 18X₁ (the second measuringinstrument) and measurement results (X₂, X₁) of one of the first and thesecond measuring instruments were synthesized via high pass filter (F₁)and low pass filter (F₂), respectively, instead of the high pass filterand the low pass filter, for example, the synthesization can beperformed a using a bandpass filter, a notch filter and the like. Thatis, as long as the structure is a structure in which syntheticcontrolled variable X_(mix) is obtained by synthesizing the frequencyband where there is a resonance mode of synthetic quantity (X_(c)) andthe frequency band where there are no resonance modes of measurementresults (X₂, X₁) of one of the first and the second measuringinstruments, any filter can be used to structure the feedback controlsystem of the SIMO system.

Further, in the embodiment above, while the case has been describedwhere driving of plate stage PST in the X-axis direction was controlled,the feedback control system can be designed similarly to the case wheredriving of plate stage PST is controlled in the Y-axis direction and theZ-axis direction, and an equivalent efficacy can be obtained.

A Second Embodiment

Next, a second embodiment of the present invention will be described,using FIGS. 13 to 17. Here, the same reference signs will be used forcomponents the same as the first embodiment previously described, and adetailed description thereabout will also be omitted.

In the feedback control system (FS-SRC) of the SIMO system previouslydescribed in the first embodiment, by designing synthetic section 52focusing on one resonance mode, robust drive control of plate stage PSTin the high bandwidth became possible without observing the resonancemode. However, in the case a plurality of resonance modes exist,resonance modes other than the resonance mode being focused on whendesigning synthetic section 52 will be observed. Therefore, in thefeedback control system (FS-SRC) of the present embodiment, syntheticsection 52 is designed divided for each frequency band in which each ofthe plurality of resonance modes exists. The feedback control systemhaving this structure in which the feedback control system (FS-SRC)previously described is expanded into the plurality of resonance modeswill be referred to as a MultiFS-SRC.

FIG. 13 shows a block diagram showing a closed loop control system(feedback control system) of a single-input two-output system (SIMOsystem) corresponding to the driving system of plate stage PST relatedto the present embodiment. Compared to the feedback control system(FS-SRC) of the SIMO system in the first embodiment, only the design ofsynthetic section 52 differs. Therefore, the description will be madeonly on design of synthetic section 52. However, there is to be aplurality of resonance modes, and of such modes, N (≧2) resonance modeswill be taken into consideration.

Synthetic section 52 includes N sets of comparators (proportional gainsβ_(n), α_(n)) 52 _(n1), 52 _(n2), adder 52 _(n3) (n=1 to N), N+1 filter52 _(n4) (n=0 to N), and one adder 52 _(m).

By the n^(th) set of comparators 52 _(n1), 52 _(n2) and adder 52 _(n3)synthesizing X position X₂ (current position) of plate table PTB(transfer function P₂) measured by interferometer 18X and X position X₁(current position) of carriage 30 (transfer function P₁) measured byinterferometer 18X₁, an intermediate synthetic quantity (X_(srcn)) isgenerated. Here, comparators (proportional gains β_(n), α_(n)) 52 _(n1),52 _(n2) multiply measurement results X₁, X₂ from interferometers 18X₁,18X by proportional gains β_(n), α_(n) times (β_(n)X₁, α_(n)X₂),respectively, and sends the results to adder 52 _(n3). Adder 52 _(n3)generates a sum (α_(n)X₂+β_(n)X₁) of the outputs from comparators 52_(n1), 52 _(n2), and supplies the sum as the intermediate syntheticquantity (X_(srcn)=α_(n)X₂+β_(n)X₁) to filter 52 _(n4). N sets ofcomparators 52 _(n1), 52 _(n2) and adder 52 _(n3) (n=1 to N) are allstructured in a similar manner.

N sets of comparators 52 _(n1), 52 _(n2) (n=1 to N) are each designed,focusing on the n^(th) resonance mode. The details are as is describedin the first embodiment, and an appropriate model that shows the n^(th)resonance mode is employed and proportional gains β_(n), α_(n) ofcomparators 52 _(n1), 52 _(n2) are decided.

N filter 52 _(n4) (n=1 to N) performs filter processing Fn (X_(srcn)) oneach input signal (intermediate synthetic quantity X_(srcn)) andsupplies the signals to adder 52 _(m). Here, passing band of filter 52_(n4) includes resonance frequency ω_(n) of the corresponding n^(th)resonance mode and the frequency band around resonance frequency ω_(n).However, the passing band of N filter 52 _(n4) (n=1 to N) is divided tokeep the passing bands from overlapping one another.

Meanwhile, X position X₂ of plate table PTB (transfer function P₂)measured by interferometer 18X is supplied to filter 52 ₀₄. Filter 52 ₀₄performs filter processing F₀(X₂) on input signal X₂, and supplies thesignal to adder 52 _(m). Here, in the passing band of filter 52 _(n4), aband other than the passing band of N filter 52 _(n4) (n=1 to N), in thepresent embodiment, a low-frequency band where there is no resonancemode, is included.

Adder 52 _(m) synthesizes signal F₀(X₂) from N+1 filter 52 _(n4) (n=0 toN) and Fn (X_(srcn)) and generates synthetic quantityX_(mix)=F₀(X₂)+Σ_(n=1 to N)F_(n)(X_(srcn)) and supplies the quantity tostage controller 50 (subtracter 50 ₂).

As a concrete example of N+1 filter 52 _(n4) (n=0 to N), a filter givenby formula (17a) below, and a filter given by formula (17b) areincluded.

$\begin{matrix}\left\{ \begin{matrix}{F_{0} = \frac{\omega_{0}}{s + \omega_{0}}} & \; \\{F_{n} = {\prod\limits_{m = 0}^{n - 1}\;{\frac{s}{s + \omega_{m}} \cdot \frac{\omega_{n}}{s + \omega_{n}}}}} & {{{for}\mspace{14mu} 1} \leq n \leq {N - 1}} \\{F_{N} = {\prod\limits_{m = 0}^{N - 1}\;\frac{s}{s + \omega_{n}}}} & \;\end{matrix} \right. & \left( {17a} \right) \\\left\{ \begin{matrix}{F_{0} = {\prod\limits_{m = 1}^{N}\; N_{m}}} \\{F_{1} = {1 - N_{1}}} \\{F_{n} = {{\prod\limits_{m = 1}^{n - 1}\;{{N_{m}\left( {1 - N_{n}} \right)}\mspace{14mu}{for}\mspace{14mu} 2}} \leq n \leq N}}\end{matrix} \right. & \left( {17b} \right)\end{matrix}$However, in formula (17b) above, function N, is a notch filter given byformula (18) below.

$\begin{matrix}{N_{n} = \frac{s^{2} + {2d_{n}\zeta_{n}\omega_{n}s} + \omega_{n}^{2}}{s^{2} + {2\zeta_{n}\omega_{n}s} + \omega_{n}^{2}}} & (18)\end{matrix}$Filter F₀ is a low pass filter which passes only the frequency bandlower than frequency f₀ (=ω₀/2π) among its input signal (X₂). FilterF_(n) (n=1 to N−1) is a bandpass filter which passes only the frequencyband higher than frequency f_(n−1)(=ω_(n−1)/2π) and lower than frequencyfn (=ω_(n)/2π) among its input signal (X_(srcn)) Filter F_(N) is a highpass filter which passes only the frequency band higher than frequencyf_(N)(=ω_(N)/2π) among its input signal (X_(srcn)).

Both filters of formulas (17a) and (17b) are decided so as to satisfycondition Σ_(n=0 to N)F_(n)=1, and the passing bands are not to overlapthe respective passing bands of N+1 filter 52 _(n4) (n=0 to N).

Synthetic quantity X_(mix) generated in the feedback control system(MultiFS-SRC) having the structure described above is the plant X₂ inthe low frequency band (ω<ω₀) where there is no resonance, is X_(srcn)in frequency band (ω_(n−1)≦ω<ω_(n)) where the n^(th) resonance modeexists, and is X_(srcN) in frequency band (ω≧ω_(N)) where the N^(th)resonance mode exists. This allows separation into each frequency bandwhere each of the plurality of resonance modes exist, and toindividually design corresponding comparators 52 _(n1), 52 _(n2) (n=1 toN) focusing on the respective resonance modes.

The inventors verified by simulation the performance of the feedbackcontrol system designed in the manner described above (MultiFS-SRC) andthe feedback control system of the first embodiment previously described(FS-SRC).

FIG. 14 shows plant characteristics P₁, P₂ of carriage 30 and platetable PTB (a Bode diagram showing frequency response characteristics,that is, a gain diagram (drawing on the upper side) and a phase diagram(drawing on the lower side)) that are subject to simulation. Here, P₁,P₂ are transfer functions P₁ (=X₁/U) P₂ (=X₂/U) expressing input-outputresponses (controlled variables X₁, X₂ to control input U (driving forceF)) of carriage 30 and plate table PTB, respectively. In plantcharacteristics P₁, P₂, a first resonance mode caused by falling ofplate table PTB to carriage 30 appears around 20 Hz, and a secondresonance mode caused by distortion of plate table PTB appears around 60Hz.

In the feedback control system (MultiFS-SRC) the control system(synthetic section 52) was designed taking into consideration both ofthe two resonance modes. Incidentally, to the first resonance mode,comparators 52 ₁₁, 52 ₁₂ in synthetic section 52 were designed(proportional gains β₁, α₁ were decided) by applying the invertedpendulum model shown in FIG. 8. Further, to the second resonance mode,because a two mass system model such as the inverted pendulum modelcannot be applied and a complex continuous body model is required,comparators 52 ₂₁, 52 ₂₂ were designed (proportional gains β₂, α₂ weredecided) by simulation. In the feedback control system (FS-SRC), thecontrol system (synthetic section 52) was designed taking intoconsideration only the second resonance mode. Similarly to the feedbackcontrol system (MultiFS-SRC), comparators 52 ₁, 52 ₂ in syntheticsection 52 were designed (proportional gains β, α were decided) bysimulation.

FIG. 15 shows a plant characteristic P₂ of plate table PTB in the casewhen the two feedback control systems (MultiFS-SRC and FS-SRC) areapplied (a Bode diagram showing the frequency response characteristics,that is, a gain diagram (drawing on the upper side) and a phase diagram(drawing on the lower side)). Because synthetic section 52 was designedtaking into consideration the second resonance mode in both of the twofeedback control systems (MultiFS-SRC and FS-SRC), the resonance mode ismade unobservable at around 60 Hz. Meanwhile, in the feedback controlsystem (MultiFS-SRC), because synthetic section 52 is designed takinginto consideration the first resonance mode, the resonance mode is madeunobservable at around 20 Hz, whereas in the feedback control system(FS-SRC), because the first resonance mode is not taken intoconsideration, the resonance mode appears at around 20 Hz.

FIG. 16 shows a sensitivity function (closed loop transfer function) inthe case when the two feedback control systems (MultiFS-SRC and FS-SRC)are applied. In both of the two feedback control systems (MultiFS-SRCand FS-SRC), the resonance mode is made unobservable at around 60 Hz.Meanwhile, in the feedback control system (MultiFS-SRC), while theresonance mode at around 20 Hz is made unobservable, in the feedbackcontrol system (FS-SRC), a peak deriving from the first resonance modeappears.

FIG. 17 shows a Nyquist diagram in the case when the two feedbackcontrol systems (MultiFS-SRC and FS-SRC) are applied. As for feedbackcontrol system (FS-SRC), it can be seen that the locus nears point (−1,0) at around point (−0.3, −0.3), deriving from the first resonance mode.Meanwhile, in the feedback control system (MultiFS-SRC), because thefirst resonance mode is also made unobservable, the locus does notapproach point (−1, 0), and stability margin is secured.

Accordingly, it has been proved that by applying the feedback controlsystem (MultiFS-SRC), the plurality of resonance modes can be madeunobservable, and higher stability can be obtained.

As is described so far, in the feedback control system (MultiFS-SRC) ofthe SIMO system in the present embodiment, a plurality of syntheticquantities (X_(cn)=α_(n)X₂+β_(n)X₁ (n=1 to N)) is obtained usingmeasurement results (X₂, X₁) of interferometer 18X (the first measuringinstrument) and interferometer 18X₁ (the second measuring instrument)and transfer functions (α_(n), β_(n) (n=1 to N)) of a plurality of sets(N (≧2) sets), and by performing filter processing on the plurality ofsynthetic quantities and measurement result (X₂) of interferometer 18X(the first measuring instrument), synthetic controlled variableX_(mix)=F₀(X₂)+Σ_(n=1 to N)F_(n) (α_(n)X₂+β_(n)X₁) is obtained. Here,comparators 52 _(n1), 52 _(n2) (n=1 to N) are designed individually(proportional gains β_(n), α_(n) are decided) so as to performseparation into each frequency band in which the plurality of resonancemodes exist and to respond focusing on the respective resonance modes.This can make the plurality of resonance modes unobservable, and adriving system that can control the driving of plate stage PST with ahigher stability can be obtained.

Incidentally, in the first and the second embodiments described above,the structure was employed where the position of the first section(plate table PTB) (the first controlled variable X₂) and the position ofthe second section (carriage 30) (the second controlled variable X₁) ofplate stage PST were measured, respectively, using interferometer 18X(the first measuring instrument) and interferometer 18X₁ (the secondmeasuring instrument) of plate interferometer system 18. Instead ofthis, for example, a structure can be employed where the first measuringinstrument measures the position of the first section (plate table PTB)with the position of the second section (carriage 30) serving as areference. On the contrary, a structure can be employed where the secondmeasuring instrument measures the position of the second section(carriage 30) with the position of the first section (plate table PTB)serving as a reference. That is, a structure can be employed where oneof the first and the second measuring instruments measures a relativeposition between the first section (plate table PTB) and the secondsection (carriage 30) of plate stage PST. In such a case, one of themeasuring instruments is not limited to the interferometer, and forexample, it is also possible to use an encoder which irradiates ameasurement beam using a head provided on one of plate table PTB andcarriage 30 onto a scale provided on the other of plate table PTB andcarriage 30 and receives the return light.

Further, the structure of plate interferometer system 18 is not limitedto the structure described above, and according to the purpose, astructure in which an interferometer is added furthermore can also beemployed, as appropriate. Further, instead of plate interferometersystem 18, or along with plate interferometer system 18, an encoder (oran encoder system structured from a plurality of encoders) can also beused.

Incidentally, each of the embodiments described above is effective inparticular in the case when the exposure object is a substrate having asize which is 500 mm or more (on the long side or in diameter).

Further, the illumination light can also be ultraviolet light such as anArF excimer laser beam (wavelength 193 nm) or a KrF excimer laser beam(wavelength 248 nm), or vacuum-ultraviolet light such as a F₂ excimerlaser beam (wavelength 157 nm). Further, as the illumination light, forexample, a harmonic wave can also be used which is a single-wavelengthlaser beam in the infrared or visible range emitted by a DFBsemiconductor laser or a fiber laser that is amplified by a fiberamplifier doped with, for example, erbium (or both erbium andytterbium), and whose wavelength is converted into ultraviolet lightusing a nonlinear optical crystal. Further, a solid-state laser(wavelength: 355 nm, 266 nm) and the like can also be used.

Further, in the embodiments described above, while the case has beendescribed where projection optical system PL is a multi-lens projectionoptical system which is equipped with a plurality of optical systems,the number of projection optical systems is not limited to this, and theprojection optical system can be one or more. Further, the projectionoptical system is not limited to the multi-lens projection opticalsystem, and for example, can also be an Offner type projection opticalsystem which uses a large mirror. Further, in the embodiments describedabove, while the case has been described where a system that has anequal magnification was used as projection optical system PL, thepresent invention is not limited to this, and the projection opticalsystem can also either be a magnifying system or a reduction system.

Further, (the stage driving system in) each of the embodiments describedabove can also be applied to a collective exposure type or a scanningtype exposure apparatus such as a scanning stepper, or to a stationarytype exposure apparatus such as a stepper. Further, each of theembodiments above can also be applied to a projection exposure apparatusthat employs a step-and-stitch method in which a shot area and a shotarea are synthesized. Further, each of the embodiments above can also beapplied to an exposure apparatus by a proximity method that does not useany projection optical systems, as well as to a liquid immersion typeexposure apparatus that exposes a substrate via an optical system and aliquid. Besides such apparatuses, each of the embodiments above can alsobe applied to an exposure apparatus such as an exposure apparatus whichsynthesizes two patterns on a substrate via a projection optical system,and by performing scanning exposure once, performs double exposure ofone shot area on the substrate almost simultaneously (U.S. Pat. No.6,611,316).

Further, the usage of the exposure apparatus is not limited to theexposure apparatus for liquid crystals to transfer a liquid crystaldisplay devices pattern on a square-shaped glass plate, and can also bewidely applied, for example, to an exposure apparatus for producingsemiconductors, or to an exposure apparatus used to manufacture thinfilm magnetic heads, micromachines, DNA chips and the like. Further,each of the embodiments above can be applied not only to an exposureapparatus for microdevices such as semiconductor devices but also to anexposure apparatus that transfers a circuit pattern on a glasssubstrate, a silicon wafer or the like to manufacture a mask or areticle used in a light exposure apparatus, an EUV exposure apparatus,an X-ray exposure apparatus, an electron beam exposure apparatus and thelike. Incidentally, the object subject to exposure is not limited to aglass plate, and can be, for example, other objects such as a wafer, aceramic substrate, a mask blank or the like.

Electronic devices such as liquid crystal display devices (orsemiconductor devices) are manufactured through the steps of; a step inwhich function/performance design of the device is performed, a step inwhich a mask (or a reticle) is manufactured based on the design step, astep in which a glass plate (or a wafer) is manufactured, a lithographystep in which the pattern of the mask (reticle) is transferred onto theglass plate by the exposure apparatus, and the exposure method describedin each of the embodiments above, a development step in which the glassplate that has been exposed is developed, an etching step in which anexposed member of apart other than the part where the resist remains isremoved by etching, a resist removing step in which the resist that isno longer necessary is removed when etching has been completed, a deviceassembly step, an inspection step, and the like. In this case, in thelithography step, because the device pattern is formed on the glassplate by executing the exposure method previously described using theexposure apparatus described above, a highly integrated device can beproduced with good productivity.

Incidentally, the disclosures of all the Publications and U.S. patentsthat are cited in the description so far related to exposure apparatusesand the like are each incorporated herein by reference.

What is claimed is:
 1. A driving system which drives a plant by givingcontrol input, the system comprising: a first measuring instrument whichmeasures a first controlled variable related to a position of a firstsection of the plant; a second measuring instrument which measures asecond controlled variable related to a position of a second section ofthe plant that shows a behavior including a resonance mode in oppositephase to a rigid-body mode that the first section shows; and acontroller which obtains a third controlled variable by obtainingsynthetic quantity (X_(c)=αX₂+βX₁) using measurement results of thefirst and the second controlled variables (X₂, X₁) by the first and thesecond measuring instruments and transfer function (α, β), and byperforming a filter processing on the synthetic quantity and one ofmeasurement results of the first and the second measuring instruments,and gives the control input obtained using the third controlled variableto the plant.
 2. The driving system according to claim 1, wherein thetransfer function (α, β) is decided so that a pole corresponding to theresonance mode included in each of transfer functions P₂, P₁corresponding to the first and the second sections are canceled out intransfer function αP₂+βP₁.
 3. The driving system according to claim 2,wherein a specific form of the transfer functions P₂, P₁ is given usinga dynamic model which expresses motion of the first and the secondsections as a motion of at least two or more rigid bodies coupled by aspring or a spring and a damper.
 4. The driving system according toclaim 1, wherein the controller obtains the third controlled variable byperforming filter processing on the synthetic quantity (X_(c)) and theone of the measurement results (X₂, X₁), and by synthesizing a frequencyband in which the resonance mode of the synthetic quantity (X_(c))exists and a frequency band in which the resonance mode of the one ofthe measurement results (X₂, X₁) is non-existent.
 5. The driving systemaccording to claim 4, wherein the controller obtains the thirdcontrolled variable (X₃=Fh(X_(c))+Fl(X₂, X₁)) by synthesizing thesynthetic quantity (X_(c)) and the one of the measurement results (X₂,X₁) via a high pass filter (Fh) and a low pass filter (Fl) having acutoff frequency the same as the high pass filter, respectively.
 6. Thedriving system according to claim 1, wherein the transfer function (α,β) is expressed by gain.
 7. An exposure apparatus that exposes an objectwith an energy beam and forms a pattern on the object, the apparatuscomprising: the driving system according to claim 1 in which a movablebody that holds the object and moves on a predetermined plane serves asthe plant.
 8. The exposure apparatus according to claim 7, wherein themovable body has a first movable body that moves holding the object, anda second movable body that moves on the predetermined plane holding thefirst movable body, and the first and the second sections of the plantare included in the first and the second movable bodies, respectively.9. A driving system which drives a plant by giving a control input, thesystem comprising: a first measuring instrument which measures a firstcontrolled variable related to a position of a first section of theplant; a second measuring instrument which measures a second controlledvariable related to a position of a second section of the plant thatshows a behavior including a resonance mode in opposite phase to arigid-body mode that the first section shows; and a controller whichobtains a third controlled variable by obtaining a plurality ofsynthetic quantities (X_(cn)=α_(n)X₂+β_(n)X₁ (n=1 to N)) usingmeasurement results of the first and the second controlled variables(X₂, X₁) by the first and the second measuring instruments and aplurality of sets (N (≧2) sets) of transfer function (α_(n), β_(n) (n=1to N)) and performing filter processing on the plurality of syntheticquantities and one of the measurement results of the first and thesecond measuring instruments, and gives the control input which isobtained using the third controlled variable to the plant.
 10. Thedriving system according to claim 9, wherein an n^(th) set transferfunction (α_(n), β_(n)) of the plurality of sets of transfer functionsis decided so that poles corresponding to an n^(th) resonance modeincluded in transfer functions P₂, P₁, respectively, corresponding tothe first and the second sections are canceled out in transfer functionα_(n)P₂+β_(n)P₁.
 11. The driving system according to claim 10, wherein aspecific form of the transfer functions P₂, P₁ is given, using a dynamicmodel which expresses a motion of the first and the second sections as amotion of at least two or more rigid bodies coupled by a spring or aspring and a damper.
 12. The driving system according to claim 10,wherein the controller obtains the third controlled variable byperforming filter processing on the plurality of synthetic quantities(X_(cn) (n=1 to N)) and the one of the measurement results (X₂, X₁), andsynthesizing a frequency band in which resonance modes corresponding toeach of the plurality of synthetic quantities (X_(cn) (n=1 to N)) existand a frequency band other than the frequency band of the one of themeasurement results (X₂, X₁).
 13. The driving system according to claim9, wherein the transfer function is expressed by gain.
 14. The drivingsystem according to claim 9, wherein one of the first and the secondmeasuring instruments measures a controlled variable with a positionsubject to measurement of an other of the first and the second measuringinstruments serving as a reference.
 15. An exposure apparatus thatexposes an object with an energy beam and forms a pattern on the object,the apparatus comprising: the driving system according to claim 9 inwhich a movable body that holds the object and moves on a predeterminedplane serves as the plant.
 16. The exposure apparatus according to claim15, wherein the movable body has a first movable body that moves holdingthe object, and a second movable body that moves on the predeterminedplane holding the first movable body, and the first and the secondsections of the plant are included in the first and the second movablebodies, respectively.
 17. An exposure apparatus that exposes an objectwith an energy beam and forms a pattern on the object, the apparatuscomprising: a movable body which has a first movable body that movesholding the object, and a second movable body that moves on apredetermined plane holding the first movable body; a first measuringinstrument that measures a first controlled variable related to aposition of the first movable body and a second measuring instrumentthat measures a second controlled variable related to a position of thesecond movable body; and a controller which obtains a third controlledvariable by obtaining an N synthetic quantity (X_(cn)=α_(n)X₂+β_(n)X₁(n=1 to N)) using measurement results of the first and the secondcontrolled variables (X₂, X₁) by the first and the second measuringinstruments and N sets (N≧1), which is one set or more, of transferfunctions (α_(n), β_(n) (n=1 to N)), and by performing a filterprocessing on the synthetic quantity and one of the measurement resultsof the first and the second measuring instruments, and drives themovable body by giving the control input obtained using the thirdcontrolled variable to the movable body.
 18. The exposure apparatusaccording to claim 17, wherein the second measuring instrument is placedat a part of the second movable body that shows a behavior including aresonance mode in opposite phase to a rigid-body mode that the firstmovable body shows.
 19. The exposure apparatus according to claim 17,wherein an n^(th) transfer function (α, β_(n)) of the N sets of transferfunctions is decided, so that poles corresponding to n^(th) resonancemodes included in transfer functions P₂, P₁, respectively, correspondingto the first and the second movable bodies are canceled out in transferfunction α_(n)P₂+β_(n)P₁.
 20. The exposure apparatus according to claim19, wherein a specific form of the transfer functions P₂, P₁ is given,using a dynamic model which expresses a motion of the first and thesecond movable bodies as a motion of at least two or more rigid bodiescoupled by a spring or a spring and a damper.
 21. The exposure apparatusaccording to claim 17, wherein the controller obtains the thirdcontrolled variable by performing filter processing on the N syntheticquantity (X_(cn) (n=1 to N)) and the one of the measurement results (X₂,X₁), and synthesizing a frequency band in which resonance modescorresponding to each of the N synthetic quantities (X_(cn) (n=1 to N))exist and a frequency band other than the frequency band of the one ofthe measurement results (X₂, X₁).
 22. The exposure apparatus accordingto claim 17, wherein the transfer function is expressed by gain.
 23. Theexposure apparatus according to claim 17, wherein one of the first andthe second measuring instruments measures a controlled variable with aposition subject to measurement of an other of the first and the secondmeasuring instruments serving as a reference.
 24. A driving method inwhich a plant is driven by giving a control input, the methodcomprising: measuring a first controlled variable related to a positionof a first section of the plant and a second controlled variable relatedto a position of a second section of the plant that shows a behaviorincluding a resonance mode in opposite phase to a rigid-body mode thatfirst section shows; and driving the plant by obtaining a thirdcontrolled variable by obtaining a synthetic quantity (X_(c)=αX₂+βX₁)using the measurement results of the first and the second controlledvariables (X², X₁) and transfer function (α, β), and performing a filterprocessing on the synthetic quantity and one of the measurement resultsof the first and the second controlled variables, and giving the controlinput obtained using the third controlled variable to the plant.
 25. Thedriving method according to claim 24, wherein the transfer function (α,β) is decided so that a pole corresponding to the resonance modeincluded in each of transfer functions P₂, P₁ corresponding to the firstand the second sections are canceled out in transfer function αP₂+βP₁.26. The driving method according to claim 25, wherein a specific form ofthe transfer functions P₂, P₁ is given, using a dynamic model whichexpresses a motion of the first and the second sections as a motion ofat least two or more rigid bodies coupled by a spring or a spring and adamper.
 27. The driving method according to claim 24, wherein in thedriving, the third controlled variable is obtained by performing filterprocessing on the synthetic quantity (X_(c)) and the one of themeasurement results (X₂, X₁) and synthesizing a frequency band in whichthe resonance mode of the synthetic quantity (X_(c)) exists and afrequency band in which the resonance mode of the one of the measurementresults (X₂, X₁) is non-existent.
 28. The driving method according toclaim 27, wherein in the driving, the third controlled variable(X₃=Fh(X_(c))+Fl(X₂, X₁)) is obtained, by synthesizing the syntheticquantity (X_(c)) which passes through a high pass filter (Fh) and theone of the measurement results (X₂, X₁) which passes through a low passfilter (Fl) having a same cutoff frequency as the high pass filter. 29.The driving method according to claim 24, wherein the transfer function(α, β) is expressed by gain.
 30. An exposure method that exposes anobject with an energy beam and forms a pattern on the object, the methodcomprising: driving a movable body that holds the object and moves on apredetermined plane as the plant by the driving method according toclaim
 24. 31. The exposure method according to claim 30, wherein themovable body has a first movable body that moves holding the object, anda second movable body that moves on the predetermined plane holding thefirst movable body, and the first and the second sections of the plantare included in the first and the second movable bodies, respectively.32. A driving method in which a plant is driven by giving a controlinput, the method comprising: measuring a first controlled variablerelated to a position of a first section of the plant and a secondcontrolled variable related to a position of a second section of theplant that shows a behavior including a resonance mode in opposite phaseto a rigid-body mode that the first section shows; and driving the plantby obtaining a third controlled variable by obtaining a plurality ofsynthetic quantities (X_(cn)=α_(n)X₂+β_(n)X₁ (n=1 to N)) usingmeasurement results of the first and the second controlled variables(X₂, X₁) and a plurality of sets (N (N (≧2) sets) of transfer function(α_(n), β_(n) (n=1 to N)), performing filter processing on the pluralityof synthetic quantities and one of the measurement results of the firstand the second controlled variables, and giving the plant the controlinput which is obtained using the third controlled variable.
 33. Thedriving method according to claim 32, wherein an n^(th) set transferfunction (α_(n), β) of the plurality of sets of transfer functions isdecided so that poles corresponding to an n^(th) resonance mode includedin transfer functions P₂, P₁, respectively, corresponding to the firstand the second sections are canceled out in transfer functionα_(n)P₂+β_(n)P₁.
 34. The driving method according to claim 33, wherein aspecific form of the transfer functions P₂, P₁ is given, using a dynamicmodel which expresses a motion of the first and the second sections as amotion of at least two or more rigid bodies coupled by a spring or aspring and a damper.
 35. The driving method according to claim 33,wherein in the driving, the third controlled variable is obtained byperforming filter processing on the plurality of synthetic quantities(X_(cn) (n=1 to N)) and the one of the measurement results (X₂, X₁), andsynthesizing a frequency band in which a resonance mode corresponding toeach of the plurality of synthetic quantities (X_(cn) (n=1 to N)) existand a frequency band other than the frequency band of the one of themeasurement results (X₂, X₁).
 36. The driving method according to claim32, wherein the transfer function is expressed by gain.
 37. The drivingmethod according to claim 32, wherein in the measuring, one of the firstand the second controlled variables is measured with an other of thefirst and the second controlled variables serving as a reference.
 38. Anexposure method that exposes an object with an energy beam and forms apattern on the object, the method comprising: driving a movable bodythat holds the object and moves on a predetermined plane as the plant bythe driving method according to claim
 32. 39. The exposure methodaccording to claim 38, wherein the movable body has a first movable bodythat moves holding the object, and a second movable body that moves onthe predetermined plane holding the first movable body, and the firstand the second sections of the plant are included in the first and thesecond movable bodies, respectively.
 40. An exposure method that exposesan object with an energy beam and forms a pattern on the object, themethod comprising: measuring a first controlled variable related to aposition of a first movable body which moves holding the object, and asecond controlled variable related to a position of a second movablebody which moves on a predetermined plane holding the first movablebody; and driving a movable body by obtaining a third controlledvariable, which is obtained by obtaining an N synthetic quantity(X_(cn)=α_(n)X₂+β_(n)X₁ (n=1 to N)) using measurement results of thefirst and the second controlled variables (X₂, X₁)) and N sets (N≧1 ),which is one set or more, of transfer functions (α_(n), β_(n) (n=1 )),and by performing a filter processing on the synthetic quantity and oneof the measurement results of the first and the second controlledvariables.
 41. The exposure method according to claim 40, wherein in themeasuring, the second controlled variable related to a position of apart of the second movable body that shows a behavior including aresonance mode in opposite phase to a rigid-body mode that the firstmovable body shows is measured.
 42. The exposure method according toclaim 40, wherein an n^(th) set transfer function (α_(n), β_(n)) of theN sets of transfer functions is decided so that poles corresponding toan n^(th) resonance mode included in transfer functions P₂, P₁,respectively, corresponding to the first and the second movable bodiesare canceled out in transfer function α_(n)P₂+β_(n)P₁.
 43. The exposuremethod according to claim 42, wherein a specific form of the transferfunctions P₂, P₁ is given, using a dynamic model which expresses amotion of the first and the second movable bodies as a motion of atleast two or more rigid bodies coupled by a spring or a spring and adamper.
 44. The exposure method according to claim 40, wherein in thedriving, the third controlled variable is obtained by performing filterprocessing on the N synthetic quantity (X_(cn) (n=1 to N)) and the oneof measurement results (X₂, X₁), and synthesizing a frequency band inwhich resonance modes corresponding to each of the N syntheticquantities (X_(cn) (n=1 to N)) exist and a frequency band other than thefrequency band of the one of the measurement results (X₂, X₁).
 45. Theexposure method according to claim 40, wherein the transfer function isexpressed by gain.
 46. The exposure method according to claim 40,wherein in the measuring, one of the first and the second controlledvariables is measured using an other of the first and the secondcontrolled variables as a reference.
 47. A device manufacturing method,comprising: forming a pattern on an object using the exposure methodaccording to claim 38, and developing the object on which the pattern isformed.
 48. A device manufacturing method, comprising: forming a patternon an object using the exposure method according to claim 46, anddeveloping the object on which the pattern is formed.